The inventor of the invention described herein is an employee of the United States Government. Therefore, the invention may be manufactured and used by or for the Government for governmental purposes without the payment of any royalties thereon or therefor.
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1. Technical Field of the Invention
This invention generally relates to a signal analysis method, apparatus and article of manufacture. The results of processing several examples of biological signals are discussed herein to show the particular utility of the invention in that field and to further demonstrate the broad applicability of the invention.
Although the present invention finds utility in processing biological signals, it is to be understood that any signal representative of a real world phenomenon such as a signal representative of a physical process including electrical, mechanical, biological, chemical, optical, geophysical or other process(es) may be analyzed and thereby more fully understood by applying the invention thereto. The real world signals to which the invention finds utility include a wide variety of real world phenomena such as the behavior of a stock market, population growth, traffic flow, etc. Furthermore, the term xe2x80x9creal world signalxe2x80x9d also includes xe2x80x9cphysical signalsxe2x80x9d representative of physical processes such as the electrical, mechanical, biological, chemical, optical, geophysical process(es) mentioned above.
Although the invention is not limited to a particular type of signal processing and includes the full range of real world data representative of processes or phenomena or combinations thereof, it is most useful when such real world signals are nonlinear and nonstationary.
2. Description of Related Art
In the parent application, several examples of geophysical data signals representative of earthquakes, ocean waves, tsunamis, ocean surface elevation and wind were processed to show the invention""s wide utility to a broad variety of signal types. The techniques disclosed therein and elaborated upon herein represent major advances in physical signal processing.
Previously, analyzing signals, particularly those having nonlinear and/or nonstationary properties, was a difficult problem confronting many industries. These industries have harnessed various computer implemented methods to process data signals measured or otherwise taken from various processes such as electrical, mechanical, optical, biological, and chemical processes. Unfortunately, previous methods have not yielded results which are physically meaningful.
Among the difficulties found in conventional systems is that representing physical processes with physical signals may present one or more of the following problems:
(a) The total data span is too short;
(b) The data are nonstationary; and
(c) The data represent nonlinear processes.
Although problems (a)-(c) are separate issues, the first two problems are related because a data section shorter than the longest time scale of a stationary process can appear to be nonstationary. Because many physical events are transient, the data representative of those events are nonstationary. For example, a transient event such as an earthquake will produce nonstationary data when measured.
Nevertheless, the nonstationary character of such data is ignored or the effects assumed to be negligible. This assumption may lead to inaccurate results and incorrect interpretation of the underlying physics as explained below.
A variety of techniques have been applied to nonlinear, nonstationary physical signals. For example, many computer implemented methods apply Fourier spectral analysis to examine the energy-frequency distribution of such signals.
Although the Fourier transform that is applied by these computer implemented methods is valid under extremely general conditions, there are some crucial restrictions: the system must be linear, and the data must be strictly periodic or stationary. If these conditions are not met, then the resulting spectrum will not make sense physically.
A common technique for meeting the linearity condition is to approximate the physical phenomena with at least one linear system. Although linear approximation is an adequate solution for some applications, many physical phenomena are highly nonlinear and do not admit a reasonably accurate linear approximation.
Furthermore, imperfect probes/sensors and numerical schemes may contaminate data representative of the phenomenon. For example, the interactions of imperfect probes with a perfect linear system can make the final data nonlinear.
Many recorded physical signals are of finite duration, nonstationary, and nonlinear because they are derived from physical processes that are nonlinear either intrinsically or through interactions with imperfect probes or numerical schemes. Under these conditions, computer implemented methods which apply Fourier spectral analysis are of limited use. For lack of alternatives, however, such methods still apply Fourier spectral analysis to process such data.
In summary, the indiscriminate use of Fourier spectral analysis in these methods and the adoption of the stationarity and linearity assumptions may give inaccurate results some of which are described below.
First, the Fourier spectrum defines uniform harmonic components globally. Therefore, the Fourier spectrum needs many additional harmonic components to simulate nonstationary data that are nonuniform globally. As a result, energy is spread over a wide frequency range.
For example, using a delta function to represent the flash of light from a lightning bolt will give a phase-locked wide white Fourier spectrum. Here, many Fourier components are added to simulate the nonstationary nature of the data in the time domain, but their existence diverts energy to a much wider frequency domain. Constrained by the conservation of energy principle, these spurious harmonics and the wide frequency spectrum cannot faithfully represent the true energy density of the lighting in the frequency and time space.
More seriously, the Fourier representation also requires the existence of negative light intensity so that the components can cancel out one another to give the final delta function representing the lightning. Thus, the Fourier components might make mathematical sense, but they often do not make physical sense when applied.
Although no physical process can be represented exactly by a delta function, some physical data such as the near field strong earthquake energy signals are of extremely short duration. Such earthquake energy signals almost approach a delta function, and they always give artificially wide Fourier spectra.
Second, Fourier spectral analysis uses a linear superposition of trigonometric functions to represent the data. Therefore, additional harmonic components are required to simulate deformed wave profiles. Such deformations, as will be shown later, are the direct consequence of nonlinear effects. Whenever the form of the data deviates from a pure sine or cosine function, the Fourier spectrum will contain harmonics.
Furthermore, both nonstationarity and nonlinearity can induce spurious harmonic components that cause unwanted energy spreading and artificial frequency smearing in the Fourier spectrum. In other words, the nonstationary, stochastic nature of biological data suffers from conventional signal processing techniques and makes the interpretation of the processed data quite difficult.
According to the above background, there is a need for a more accurate signal processing technique that produces results that are more physically meaningful and readily understood. Biological signals provide another example of physical signals in which this invention is applicable. Parent application Ser. No. 08/872,586 filed Jun. 10, 1997 illustrates several other types of signals in which this invention is applicable. Namely, the parent application provides specific examples of nonlinear, nonstationary geophysical signals which are very difficult to analyze with traditional computer implemented techniques including earthquake signals, water wave signals, tsunami signals, ocean altitude and ocean circulation signals.
Many of the aforementioned signal processing problems exist when biological signals are processed. For example, most data in the field of biology are nonstationarily stochastic. When conventional tools such as Fourier Analysis are applied to such biological data, the result often-times obscures the underlying processes. In other words, conventional Fourier analysis of biological data throws away or otherwise obscures valuable information. Thus, the complex biological phenomena producing such data cannot be readily understood and is, in any event, represented imprecisely. The interpretation of the results of such conventionally processed data may, therefore, be quite difficult. The conventional techniques also make accurate modelling of the biological phenomena very difficult and, sometimes, impossible.
An object of the present invention is to solve the above-mentioned problems in conventional signal analysis techniques.
Another object of the present invention is to provide further examples of physical signal processing thereby further demonstrating the broad applicability of the invention to a wide array of physical signals which include biological signals.
Another object is to provide a technique of distilling a physical signal to the point at which the signal can be represented with an analytic function.
To achieve these objects, the invention employs a computer implemented Empirical Mode Decomposition method which decomposes physical signals representative of a physical phenomenon into components. These components are designated as Intrinsic Mode Functions (IMFs) and are indicative of intrinsic oscillatory modes in the physical phenomenon.
Contrary to almost all the previous methods, this new computer implemented method is intuitive, direct, a posteriori, and adaptive, with the basis of the decomposition based on and derived from the physical signal. The bases so derived have no close analytic expressions, and they can only be numerically approximated in a specially programmed computer by utilizing the inventive methods disclosed herein.
More specifically, the general method of the invention includes two main components or steps to analyze the physical signal without suffering the problems associated with computer implemented Fourier analysis, namely inaccurate interpretation of the underlying physics or biology caused in part by energy spreading and frequency smearing in the Fourier spectrum.
The first step is to process the data with the Empirical Mode Decomposition (EMD) method, with which the data are decomposed into a number of Intrinsic Mode Function (IMF) components. In this way, the signal will be expanded by using a basis that is adaptively derived from the signal itself.
The second step of the general method of the present invention is to apply the Hilbert Transform to the decomposed IMF""s and construct an energy-frequency-time distribution, designated as the Hilbert Spectrum, from which occurrence of physical events at corresponding times (time localities) will be preserved. There is also no close analytic form for the Hilbert Spectrum. As explained below, the invention avoids this problem by storing numerical approximations in the specially programmed computer by utilizing the inventive method.
The invention also utilizes instantaneous frequency and energy to analyze the physical phenomenon rather than the global frequency and energy utilized by computer implemented Fourier spectral analysis.
Furthermore, a computer implementing the invention, e.g., via executing a program in software, to decompose physical signals into intrinsic mode functions with EMD and generate a Hilbert spectrum is also disclosed. Because of the lack of close form analytic expression of either the basis functions and the final Hilbert spectrum; computer, implementation of the inventive methods is an important part of the overall method.
Still further, the invention may take the form of an article of manufacture. More specifically, the article of manufacture is a computer-usable medium, including a computer-readable program code embodied therein wherein the computer-readable code causes a computer to execute the inventive method.
Once the IMF""s are generated, the invention can then produce a distilled or otherwise filtered version of the original physical signal. This distillation process eliminates undesired IMF""s and thereby generates a filtered signal from which it is possible to perform a curve fitting process. In this way, it is possible to arrive at an analytic function which accurately represents the physically important components of the original signal.
Further scope of applicability of the present invention will become apparent from the detailed description given hereinafter. However, it should be understood that the detailed description and specific examples, while indicating preferred embodiments of the invention, are given by way of illustration only, since various changes and modifications within the spirit and scope of the invention will become apparent to those skilled in the art from this detailed description. Furthermore, all the mathematic expressions are used as a short hand to express the inventive ideas clearly and are not limitative of the claimed invention.
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